# Wavefunction of an electron after a slit

To properly understand the double slit experiment with electrons, you need to know what is the wavefunction of the electron after it passes through a slit.

This wavefunction cannot be a plane wave. Indeed for a single slit experiment we would have a uniform impact on the screen which is not what is observed (we should observe something like diffraction through single slit for light).

Thus: what is the wave function of an electron after it has passed through a slit, given that it was a plane wave before for example?

• The transverse momentum components are now the Fourier transform of the slit. Oct 7 '19 at 20:39
• Are you trying to explain the bright and dark areas of the diffraction pattern? Oct 7 '19 at 23:20

It will keep it's frequency and wave-number (or wave-length if $$\lambda$$ makes you happier than $$k$$), but the direction of the wave-vector will, of course, vary in space.
• To be sure to understand, it is a wave like $\frac{e^{i \vec{k}.\vec{r}}}{\sqrt{r}}$, but $\vec{k}$ being in a plane to make it cylindric ? If not could you write your reasonable approximation solution ? Oct 7 '19 at 20:16
• Using $\vec{r} = (\rho,\phi,z)$ for coordinates we have wave character given by $\psi \propto \exp \left[ i (k\hat{\rho}) \cdot \vec{r} \right] = \exp ( i k \rho )$. I think your distance dependence is right, but it still remains to stick in the angular amplitude dependence from the slit width (though you can ignore that at first if the slit separation is large compared to their width). Oct 7 '19 at 20:48